How to have a color transition material on a mesh?
Based on  Diogo Valadares's comment, you can indeed use a fresnel node + color ramp to achieve this effect. Below, I attach a fresnel node to a color ramp with a mapping node in between, giving me a transition between red and blue from the inside to the outside. 



Depending on what kind of transition you want, the fresnel node may not be perfect (you can see that on the very edge, the transition is a bit too sharp), but by playing around with the constants in the mapping node you can get a sharper or smoother transition.